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To cope with the negative oil futures price caused by the COVID-19 recession, global commodity futures exchanges temporarily switched the option model from Black--Scholes to Bachelier in 2020. This study reviews the literature on Bachelier's pioneering option pricing model and summarizes the practical results on volatility conversion, risk management, stochastic volatility, and barrier options pricing to facilitate the model transition. In particular, using the displaced Black-Scholes model as a model family with the Black-Scholes and Bachelier models as special cases, we not only connect the two models but also present a continuous spectrum of model choices.

We investigate a one-period portfolio optimization problem of a cumulative prospect theory (CPT) investor with multiple risky assets and one risk-free asset. The returns of the multiple risky asset...

In this article we demonstrate the practical application of standard partial equilibrium, in backward stochastic differential equation (BSDE) framework, to pricing longevity bonds which are essential for functioning of the life market. The market for mortality linked instruments or so-called life market, is experiencing steady progress in its evolution (see Blake et al. (2018)), however, it is regretfully far from reaching its full potential estimated to be of tens of trillions of dollars (see Michaelson and Mulholland (2014)). Highly illiquid and, compared to equity markets, with a relatively very low number of transactions, the life market is comprised of series of negotiated, high monetary value, over-the-counter transactions between few agents that have different risk preferences. To accommodate these realities we demonstrate an application of a partial equilibrium framework for pricing longevity bonds. We do this under the assumption of stochastic mortality intensity that affects the income of economic agents who trade in risky financial security and longevity bond to maximize their monetary utilities. Thus, by rooting ourselves in a foundational economic principle, as a practical contribution, we find the endogenous equilibrium bond price which is numerically computed. In a realistic setting of two agents in a transaction, numerical experiments confirm the expected intuition of price dependence on model parameters. As a theoretical contribution, we prove that our longevity bond completes the market.

We investigate an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time with constant labor wage and disutility. We allow the preference of the agent to be changed before and after retirement. It is assumed that the agent's coefficient of relative risk aversion becomes higher after retirement. Under a constant relative risk aversion (CRRA) utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods. We give some numerical results of the optimal policies. We also consider the relation between the level of disutility and the labor wage with the optimal retirement wealth level.

We propose an equilibrium pricing model in a dynamic multiperiod stochastic framework with uncertain income. There are one tradable risky asset (stock/commodity), one nontradable underlying (temperature), and also a contingent claim (weather derivative) written on the tradable risky asset and the nontradable underlying in the market. The price of the contingent claim is priced in equilibrium by optimal strategies of representative agent and market clearing condition. The risk preferences are of exponential type with a stochastic coefficient of risk aversion. Both subgame perfect strategy and naive strategy are considered and the corresponding equilibrium prices are derived. From the numerical result we examine how the equilibrium prices vary in response to changes in model parameters and highlight the importance of our equilibrium pricing principle.

We investigate a one-period portfolio optimization problem of a cumulative prospect theory (CPT) investor with multiple risky assets and one risk-free asset. The returns of multiple risky assets follow multivariate generalized hyperbolic (GH) skewed t distribution. We obtain a three-fund separation result of two risky portfolios and risk-free asset. Furthermore, we reduce the high dimensional optimization problem to two 1-dimensional optimization problems and derive the optimal portfolio. We show that the optimal portfolio composition changes as some of investor-specific parameters change. It is observed that the consideration of skewness of stock return distribution has considerable impact on the distribution of CPT investor's wealth deviation, and leads to less total risky investment.

In this paper, we propose an equilibrium pricing model in a dynamic multi-period stochastic framework with uncertain income. There are one tradable risky asset (stock/commodity), one non-tradable underlying (temperature), and also a contingent claim (weather derivative) written on the tradable risky asset and the non-tradable underlying in the market. The price of the contingent claim is priced in equilibrium by optimal strategies of representative agent and market clearing condition. The risk preferences are of exponential (CARA) type with a stochastic coefficient of risk aversion. Both Nash subgame perfect strategies and naive strategies are considered. From the numerical result we examine how the equilibrium prices vary in response to changes in model parameters and highlight the importance of Nash equilibrium pricing principle.

This paper investigates a continuous-time optimal consumption, investment, and life insurance decision problem of a family under inflation risk. In the financial market, there is a liquid inflation-linked index bond market which can be utilized to hedge the inflation risk. The explicit solutions are derived for constant relative risk aversion (CRRA) utility case by using martingale approach. The roles of index bond are investigated and it is verified that the index bond may have different roles depending on the market parameters. We analyze the effects of parameters on the optimal strategies with focus on the optimal demand for index bond and the optimal life insurance premium. Especially, the changes of expected inflation rate and volatility of inflation rate can have both positive and negative impacts on the life insurance premium and their quantitative impacts are considerable.